2012/09/25

Way to rationalize

I confess. I claimed Gauche conformed R5RS, but it didn't. In the early stage of development, I somehow thought rationalize was an optional procedure and I didn't bother to implement it. But it wasn't optional; it was a library procedure. Ugh.

So one day I decided to write one, and here's my little journey.

What (rationalize x e) returns is the simplest rational number p/q, where |x - p/q| is within the specified error bound e.

A quick search finds a method to use Stern-Brocot tree. Suppose we have a/b and c/d where a/b < x < c/d. We calculate the mediant M = (a+c)/(b+d), then we have the following cases:

• If M is close enough to x, it's the answer.
• If M is above x, shrink the search range to a/b < x < M and recurse.
• If M is below x, shrink the search range to M < x < c/d and recurse.

Reasonable. This is the straight implementation (for simplicity, I made it always return an exact rational, and also omitted handling exceptional cases like +inf.0).

Looks good, yeah?

gosh> (use math.const)
#<undef>
gosh> (rationalize1 pi 1/100)
22/7
gosh> (rationalize1 pi 1/1000)
201/64
gosh> (rationalize1 pi 1/100000)
355/113
gosh> (rationalize1 pi 1/10000000)
75948/24175
gosh> (rationalize1 (/ 3.0) 1e-17)
1/3

Well, not quite. This doesn't work well when the error bound is small and x falls just outside of error bound from either a/b or c/d. Suppose x is 1/1000000 and error bound is 1/2000000. We start from 0/1 and 1/1, but the mediant is always greater than x so only higher bound is moving towards x. The denominator of the lower bound is fixed to 1, so the mediant's denominator increases only 1 at every iteration. It'll take 1000000 steps to get the answer.

So I searched a bit more. Someone described an algorithm about calculating (regular) continued fraction up to the point the error gets small enough. Calculating continued fraction is easy; it's basically the same as Euclidean algorithm.

gosh> (continued-fraction (exact pi))
(3 7 15 1 292 1 1 1 2 1 3 1 14 3 3 2 1 3 3 7 2 1 1 3 2 42 2)
gosh> (continued-fraction (exact e))
(2 1 2 1 1 4 1 1 6 1 1 8 1 1 10 1 1 12 1 1 11 1 1 1 11 5 1 1 2 1 4 2 1 1 9 17 3)

That is, π = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...)))) and so on.

(Note1: Since we have limited precision in the floating number constant pi and e, the above expansions don't match the real expansion of pi and e after first dozen items or so.)

(Note2: Each rational number has two regular continued fraction representation, for you can always decrement the last item and append '1' to the tail; e.g. 3/8 can be [0; 2, 1, 2] and [0; 2, 1, 1, 1]. Our continued-fraction procedure always returns the shorter one.)

Suppose we have a continued fraction [a0; a1, a2, ...], and its calculated value up to n-th term is p_n/q_n, then it is known that the value up to n+1-th term p_{n+1}/q_{n+1} is (a_{n+1}*p_{n} + p_{n-1})/(a_{n+1}*q_{n} + q_{n-1}), where p_0/q_0 = a0 and p_1/q_1 = a0 + 1/a1. So we can keep two previous approximated value to calcuate the next approximation, instead of calculating the continued fraction every time we add the term.

Here's an implementation. Again, we always return an exact result for the simplicity. Note that continued-fraction returns a lazy sequence, so we don't calculate the continued fraction more than necessary.

(If an gets empty, Qn must be exactly the same as x, so we never need to recurse further.)

Is this good enough? Unfortunately, this converges too quickly and possibly misses simpler (but less closer) approximation. In the following example, rationalize1 finds the desired approximation (1/500) but rationalize2 returns closer but less simple rational (1/999).

gosh> (rationalize1 2/1999 1/1000)
1/500
gosh> (rationalize2 2/1999 1/1000)
1/999

The Wikipedia entry of continued fraction has some good explanation. Basically, if we have a truncated continued fraction [a0; a1, a2, ..., an-1, an], then we have to check [a0; a1, a2, ... , an-1, ceiling(an/2)], [a0; a1, a2, ... , an-1, ceiling(an/2)+1], ... etc. This would make convergence slow again, when we have very large an.

The solution is in the next headings of the same Wikipedia article. We can find the simplest rational between two numbers, by comparing continued fractions of the two. Let's try some examples.

gosh> (continued-fraction 31415/10000)
(3 7 14 1 8 2)
gosh> (continued-fraction 31416/10000)
(3 7 16 11)

To find a simplest rational between 31415/10000 and 31416/10000, we look for the first item that differ between the two sequences---in this case, it's the third item, 14 vs 16. We replace the item with (+ (min 14 16) 1), which is 15, and discard the rest; that is, [3; 7, 15] is the answer.

gosh> (+ 3 (/ (+ 7 (/ 15))))
333/106

Given x and e, the answer of rationalize is the simplest rational between x-e and x+e, hence we get this implementation:

The actual code in Gauche includes handling of inexactness, infinites and NaNs.

(Note: The above implementation finds the answer in the closed range between x-e and x+e, which is ok for rationalize, but if you want to search in the open range, the refine part will be a bit more complicated. Besides, the above implementation doesn't handle the case when e >= 1.)

Tags: r5rs, rationalize